#6477172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,32,42)(20,33,21,25)(22,36,38,26)(24,37,40,31)(29,44,30,43)(34,41,35,48)(39,45,47,46);(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,27,40)(20,29,21,30)(22,37,38,31)(25,39,33,47)(26,42,36,28)(34,46,35,45)(41,44,48,43);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,23,21)(18,25,27,33)(19,29,32,30)(22,34,38,35)(24,39,40,47)(26,41,36,48)(28,43,42,44)(31,45,37,46);(11,15)(13,16)(17,21,23,20)(18,25,27,33)(19,30,32,29)(22,35,38,34)(24,39,40,47)(26,41,36,48)(28,43,42,44)(31,46,37,45);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,21)(25,33)(29,30)(34,35)(39,47)(41,48)(43,44)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384368);